An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.

The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of an object language. For example, an interpretation function could take the predicate symbol ${\displaystyle T}$ and assign it the extension ${\displaystyle \{(\mathrm {a} )\}}$. All our interpretation does is assign the extension ${\displaystyle \{(\mathrm {a} )\}}$ to the non-logical symbol ${\displaystyle T}$, and does not make a claim about whether ${\displaystyle T}$ is to stand for tall and ${\displaystyle \mathrm {a} }$ for Abraham Lincoln. On the other hand, an interpretation does not have anything to say about logical symbols, e.g. logical connectives "${\displaystyle \mathrm {and} }$", "${\displaystyle \mathrm {or} }$" and "${\displaystyle \mathrm {not} }$". Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.

An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.

A formal language consists of a possibly infinite set of sentences (variously called words or formulas) built from a fixed set of letters or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined. To distinguish the strings of symbols that are in a formal language from arbitrary strings of symbols, the former are sometimes called well-formed formulæ (wff). The essential feature of a formal language is that its syntax can be defined without reference to interpretation. For example, we can determine that (P or Q) is a well-formed formula even without knowing whether it is true or false.

A formal language ${\displaystyle {\mathcal {W}}}$ can be defined with the alphabet ${\displaystyle \alpha =\{\triangle ,\square \}}$, and with a word being in ${\displaystyle {\mathcal {W}}}$ if it begins with ${\displaystyle \triangle }$ and is composed solely of the symbols ${\displaystyle \triangle }$ and ${\displaystyle \square }$.

A possible interpretation of ${\displaystyle {\mathcal {W}}}$ could assign the decimal digit '1' to ${\displaystyle \triangle }$ and '0' to ${\displaystyle \square }$. Then ${\displaystyle \triangle \square \triangle }$ would denote 101 under this interpretation of ${\displaystyle {\mathcal {W}}}$.

In the specific cases of propositional logic and predicate logic, the formal languages considered have alphabets that are divided into two sets: the logical symbols (logical constants) and the non-logical symbols. The idea behind this terminology is that logical symbols have the same meaning regardless of the subject matter being studied, while non-logical symbols change in meaning depending on the area of investigation.

Logical constants are always given the same meaning by every interpretation of the standard kind, so that only the meanings of the non-logical symbols are changed. Logical constants include quantifier symbols ∀ ("all") and ∃ ("some"), symbols for logical connectives ∧ ("and"), ∨ ("or"), ¬ ("not"), parentheses and other grouping symbols, and (in many treatments) the equality symbol =.